A Generalized Alternating Direction Method of Multipliers with Semi-Proximal Terms for Convex Composite Conic Programming

TL;DR

This paper introduces a generalized ADMM with semi-proximal terms and relaxation for convex composite conic problems, demonstrating improved performance and success on high-dimensional semidefinite programming tasks.

Contribution

It develops a flexible ADMM variant with semi-proximal and relaxation features, enhancing applicability and efficiency for complex convex conic optimization problems.

Findings

Successfully solves high-dimensional semidefinite problems

Shows the effectiveness of relaxation steps in ADMM

Demonstrates improved performance over classic ADMM

Abstract

In this paper, we propose a generalized alternating direction method of multipliers (ADMM) with semi-proximal terms for solving a class of convex composite conic optimization problems, of which some are high-dimensional, to moderate accuracy. Our primary motivation is that this method, together with properly chosen semi-proximal terms, such as those generated by the recent advance of symmetric Gauss-Seidel technique, is applicable to tackling these problems. Moreover, the proposed method, which relaxes both the primal and the dual variables in a natural way with one relaxation factor in the interval $(0,2)$, has the potential of enhancing the performance of the classic ADMM. Extensive numerical experiments on various doubly non-negative semidefinite programming problems, with or without inequality constraints, are conducted. The corresponding results showed that all these multi-block problems can be successively solved, and the advantage of using the relaxation step is apparent.